lesson applying gcf and lcm to fraction operations 4-1

Name ________________________________________ Date __________________ Class __________________

Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor.

52

Applying GCF and LCM to Fraction Operations Practice and Problem Solving: A/B

Multiply. Use the greatest common factor to write each answer in simplest form.

1. 23

i 67

2. 34

i 23

3. 821

i 710

________________________ _______________________ ________________________

4. 24 i 56

5. 32 i 38

6. 21 i 37

________________________ _______________________ ________________________

Add or subtract. Use the least common multiple as the denominator.

7. 415

+ 56

8. 512

− 320

9. 35

+ 320

________________________ _______________________ ________________________

10. 58

− 524

11. 3 512

+ 1 38

12. 2 910

− 1 718

________________________ _______________________ ________________________

Solve.

13. Louis spent 12 hours last week practicing guitar. If 14

of the time

was spent practicing chords, how much time did Louis spend practicing chords?

___________________________

14. Angie and her friends ate 34

of a pizza. Her brother Joe ate 23

of what

was left. How much of the original pizza did Joe eat?

___________________________

LESSON

4-1

Name ________________________________________ Date __________________ Class __________________


53

Applying GCF and LCM to Fraction Operations Practice and Problem Solving: C

Multiply. Use the greatest common factor to write each answer in simplest form.

1. 49

i 38

2. 79

i 314

3. 18 i 79

________________________ _______________________ ________________________

Add or subtract. Use the least common multiple as the denominator.

4. 715

+ 56

5. 1 712

− 320

6. 25

+ 720

________________________ _______________________ ________________________

Solve. 7. A recipe calls for the following ingredients.

3 c flour ______ 12

t salt ______

34

c sugar ______ 4 c fruit ______

2 T butter ______

To make 23

of the recipe, how much of each ingredient should you

use? Write the revised amount on the line next to each ingredient.

8. Half of a pizza was broccoli and half was mushroom. George ate 23

of the broccoli part and 14

of the mushroom part. How much of the

pizza did he eat?

___________________________

9. What else could you call the least common multiple in an addition or subtraction problem involving fractions?

_________________________________________________________________

LESSON

4-1

Name ________________________________________ Date __________________ Class __________________


54

Applying GCF and LCM to Fraction Operations Practice and Problem Solving: D

Multiply. Use the greatest common factor to write each answer in simplest form. The first one is done for you.

1. 23

i 34

2. 1215

i 34

3. 24 i 58

________________________ _______________________ ________________________

Add or subtract. Use the least common multiple as the denominator. The first one is done for you.

4. 1112

− 320

5. 35

+ 920

6. 1115

+ 23

________________________ _______________________ ________________________

Solve. The first one is done for you. 7. Lyza used 24 ounces of spaghetti to make a recipe. If she wanted

to make 14

as much, how much spaghetti should she use?

___________________________

8. Noah spent 25 hours working on his car. He spent 45

of his

time working on the transmission. How much time did Noah spend working on the transmission?

___________________________

9. Miguel made lemonade. He mixed 13

cup of lemon juice with

49

cup of water. How much more water than lemon juice did Miguel

use?

______________________________________________________________ ___________________________

LESSON

4-1

23 i 3

4 612 1

2

14 i 24 6; 6 ounces

Name ________________________________________ Date __________________ Class __________________


55

Applying GCF and LCM to Fraction Operations Reteach

How to Multiply a Fraction by a Fraction

23

i 38

23

i 38

6 Multiply numerators.

23

i 38

624

Multiply denominators.

6 624 6

14

Divide by the greatest common factor (GCF).

The GCF of 6 and 24 is 6. How to Add or Subtract Fractions

56

+ 1115

2530

+ 2230

Rewrite over the least common multiple (LCM).

The least common multiple of 6 and 15 is 30. 2530

+ 2230

4730

Add the numerators.

1 1730

If the sum is an improper fraction, rewrite

it as a mixed number.

Multiply. Use the greatest common factor.

1. 34

i 79

2. 27

i 79

3. 711

i 2228

________________________ _______________________ ________________________

4. 8 i 310

5. 49

i 34

6. 37

i 23

________________________ _______________________ ________________________

Add or subtract. Use the least common multiple.

7. 79

+ 512

8. 2124

− 38

9. 1115

+ 712

________________________ _______________________ ________________________

LESSON

4-1

Name ________________________________________ Date __________________ Class __________________


56

Applying GCF and LCM to Fraction Operations Reading Strategies: Use Graphic Aids

You can find the answer to 6 i 23

using fraction strips and multiplication.

6 i 23

123

1. What fractional part of each fraction strip is shaded? _______________

2. How many of these fraction strips are there? _______________

3. Write a multiplication equation for this picture. _______________

You can use fraction strips to find the least common multiple.

34

+ 13

34

+ 13

912

+ 412

So, 34

+ 13

1312

1 1 .12

Find the products. Use the greatest common factor. Write each answer in simplest form.

4. 6 i 19

5. 3 i 59

6. 2 i 1114

________________________ _______________________ ________________________

Find the sums. Use the least common multiple. Write each answer in simplest form.

7. 14

+ 512

8. 29

+ 712

9. 910

+ 14

________________________ _______________________ ________________________

LESSON

4-1

Name ________________________________________ Date __________________ Class __________________


57

Applying GCF and LCM to Fraction Operations Success for English Learners

Problem 1

Problem 2 24

+ 23

24

+ 23

612

+ 812

1412

24

+ 23

1412

76

1. Compare the steps you take to multiply fractions to the steps you take to add fractions.

_________________________________________________________________________________________

_________________________________________________________________________________________

_________________________________________________________________________________________

2. Explain how using the greatest common factor and the least common multiple helps you to multiply, and add or subtract fractions.

_________________________________________________________________________________________

_________________________________________________________________________________________

LESSON

4-1

31

i 19

39

3 39 3

13

The GCF of 3 and 9 is 3.

The LCM of 4 and 3 is 12.

THINK: What is the LCM

of 4 and 3?

2 of 4 parts

2 of 3 parts

Name ________________________________________ Date __________________ Class __________________


58

Dividing Fractions Practice and Problem Solving: A/B

Find the reciprocal.

1. 57

_______________ 2. 34

_______________ 3. 35

_______________

4. 110

_______________ 5. 49

_______________ 6. 1314

_______________

7. 712

_______________ 8. 310

_______________ 9. 58

_______________

Divide. Write each answer in simplest form.

10. 56

12

_______________ 11. 78

23

_______________ 12. 910

34

_______________

13. 34

9 _______________ 14. 69

67

_______________ 15. 56

310

_______________

16. 56

34

_______________ 17. 58

35

_______________ 18. 2132

78

_______________

Solve.

19. Mrs. Marks has 34

pound of cheese to use making sandwiches.

She uses about 132

pound of cheese on each sandwich. How many

sandwiches can she make with the cheese she has?

_____________________________________

20. In England, mass is measured in units called stones. One pound

equals 114

of a stone. A cat weighs 34

stone. How many pounds does

the cat weigh?

_____________________________________

21. Typographers measure font sizes in units called points. One point is

equal to 172

inch. Esmeralda is typing a research paper on her

computer. She wants the text on the title page to be 12

inch tall. What

font size should she use?

_____________________________________

LESSON

4-2

Name ________________________________________ Date __________________ Class __________________


59

Dividing Fractions Practice and Problem Solving: C

Find the reciprocal. Tell whether it is greater or less than 1.

1. 37

2. 34

3. 85

________________________ _______________________ ________________________

4. 111

5. 89

6. 134

________________________ _______________________ ________________________

7. If a fraction is less than 1, what do you know about its reciprocal?

_________________________________________________________________________________________

8. If a fraction is greater than 1, what do you know about its reciprocal?

_________________________________________________________________________________________

9. What is the product of a number and its reciprocal? _________________


10. 56

23

____________ 11. 78

35

____________ 12. 89

25

____________

13. 23

45

____________ 14. 57

79

____________ 15. 35

911

____________

Answer each question. 16. In problems 10–12, the dividend is greater than the divisor. What do

you know about the quotients?

_________________________________________________________________________________________

17. In questions 13–15, the divisor is greater than the dividend. What do you know about the quotients?

_________________________________________________________________________________________

18. Jonathan has 314

hours to practice guitar. If he spends 18

hour on

each song, how many songs can Jonathan practice? For how many minutes does he practice each song?

_________________________________________________________________________________________

LESSON

4-2

Name ________________________________________ Date __________________ Class __________________


60

Dividing Fractions Practice and Problem Solving: D

Find the reciprocal. The first one is done for you.

1. 23

____________ 2. 79

____________ 3. 85

____________

4. 19

____________ 5. 910

____________ 6. 310

____________

7. 47

____________ 8. 81

____________ 9. 67

____________

Divide. Write each answer in simplest form. The first one is done for you.

10. 34

12

11. 710

23

12. 56

34

________________________ _______________________ ________________________

13. 310

56

14. 59

57

15. 710

56

________________________ _______________________ ________________________

16. 78

34

17. 1112

23

18. 57

1013

________________________ _______________________ ________________________

Solve. The first one has been started for you. 19. Each package of dried fruit contains 3

16 of a pound. Mr. Lopez has

4 pounds of dried fruit. How many packages can he fill?

4 316

4 ______

3

______ ________ packages

20. One inch is 112

of a foot. Eunice has a puppy that is 34

of a foot tall.

How many inches tall is her puppy?

_________________________________________________________________________________________

21. One minute is 160

of an hour. What part of an hour is 12 minutes?

_________________________________________________________________________________________

LESSON

4-2

32

3 2 6 1 14 1 4 2i

Name ________________________________________ Date __________________ Class __________________


61

Dividing Fractions Reteach

Two numbers are reciprocals if their product is 1.

23

and 32

are reciprocals because 23

i 32

66

1.

Dividing by a number is the same as multiplying by its reciprocal.

14

12

12

14

i 21

12

So, you can use reciprocals to divide by fractions.

Find 23

14

.

First, rewrite the expression as a multiplication expression.

Use the reciprocal of the divisor: 14

i 41

1.

23

14

23

i 41

83

223

Rewrite each division expression as a multiplication expression. Then find the value of the expression. Write each answer in simplest form.

1. 14

13

2. 12

14

3. 38

12

4. 13

34

________________ ________________ _______________ ________________


5. 15

12

6. 16

23

7. 18

25

8. 18

12

________________ ________________ _______________ ________________

LESSON

4-2

Think: 6 thirds is 2, and 2 of the 8 thirds are left over.

Name ________________________________________ Date __________________ Class __________________


62

Dividing Fractions Reading Strategies: Use Models

Bar models can help you picture dividing by fractions.

What is 78

14

? Think: How many one-fourths are in 78

?

Use the picture to answer each question.

1. How many whole groups of 14

are in 78

? ___________

What fraction of a group of 14

is left? __________

2. 78

14

_________________

Instead of dividing, multiply by the reciprocal. Think: 78

four times.

Use the picture to answer each question. 3. How many whole bars are shaded? _________________

4. How many additional eighths of a bar are shaded? _________________

What is this fraction in simplest form? _________________

5. All together, how many bars are shaded? _________________

6. Compare the multiplication and division examples. What do you notice

about the answer you get when you divide by 14

or multiply by 4?

_________________________________________________________________________________________

LESSON

4-2

Name ________________________________________ Date __________________ Class __________________


63

Problem 1

Problem 2

Dividing Fractions Success for English Learners

1. How do you find the reciprocal of a fraction?

_________________________________________________________________________________________

_________________________________________________________________________________________

2. Explain the steps you follow to divide 58

by 13

.

_________________________________________________________________________________________

_________________________________________________________________________________________

3. You multiply any fraction times its reciprocal. What is the product? Give an example.

_________________________________________________________________________________________

_________________________________________________________________________________________

LESSON

4-2

Name ________________________________________ Date __________________ Class __________________


64

Dividing Mixed Numbers Practice and Problem Solving: A/B

Find the reciprocal. Show that the product of the mixed number and its reciprocal is 1.

1. 1102

2. 367

3. 829

________________________ _______________________ ________________________

4. 1154

5. 293

6. 578

________________________ _______________________ ________________________


7. 8 5110 6

8. 62 17

9. 3 13 25 4

________________________ _______________________ ________________________

10. 1 34 22 8

11. 5 15 36 6

12. 11 5212 8

________________________ _______________________ ________________________

13. 9 3113 8

14. 4 26 35 9

15. 2 89 63 9

________________________ _______________________ ________________________

Write each situation as a division problem. Then solve.

16. A concrete patio is 5 23

feet wide. It has an area of 36 56

square feet.

Is the concrete slab long enough to fit a 7-foot picnic table without placing the table along the diagonal of the patio? Explain.

_________________________________________________________________________________________

17. The area of a mirror is 225 square inches, and its width is 3134

inches.

Will the mirror fit in a space that is 15 inches by 16 inches? Explain.

_________________________________________________________________________________________

18. Barney has 1165

yards of fabric. To make an elf costume, he needs

255

yards of fabric. How many costumes can Barney make?

_________________________________________________________________________________________

LESSON

4-3

Name ________________________________________ Date __________________ Class __________________


65

Dividing Mixed Numbers Practice and Problem Solving: C

Solve. 1. Vanessa buys a strip of 25 postage stamps. The strip of stamps is

21 78

inches long. How long is a strip after Vanessa uses 1 stamp?

_________________________________________________________________________________________

2. Hasan has 3184

yards of fabric. It takes 3 16

yards to make a

pillowcase. Hasan plans to make as many pillowcases as he can. How many yards of fabric will be left over?

_________________________________________________________________________________________

3. Takafumi is hiking on a path that is 5 78

miles long. There are

6 markers evenly posted along the path. Takafumi arrives at the 4th marker. How many miles has he hiked so far?

_________________________________________________________________________________________

4. Yuki has a ribbon that is 11 14

feet long. She divides it into pieces that

are each 1 78

feet long. She uses three pieces to make a bow. How

many bows can she make in all?

_________________________________________________________________________________________

5. Mrs. Lemke has 2103

ounces of fertilizer for her plants. She plans to

use 34

ounce of fertilizer on each plant. After she puts fertilizer on as

many plants as she can, how much fertilizer will be left over?

_________________________________________________________________________________________

6. Gabriel has 15 58

pounds of clay. He will use 710

pound to make each

bowl. After making 8 bowls, Gabriel wonders how many more bowls he can make. How many more bowls can he make? Explain how you know.

_________________________________________________________________________________________

_________________________________________________________________________________________

_________________________________________________________________________________________

LESSON

4-3

Name ________________________________________ Date __________________ Class __________________


66

Dividing Mixed Numbers Practice and Problem Solving: D

Show how to write each mixed number as an improper fraction. Then find the reciprocal. The first one is done for you.

1. 192

2. 357

3. 819

________________________ ________________________ ________________________

________________________ ________________________ ________________________

4. 1144

5. 283

6. 568

________________________ _______________________ ________________________

________________________ _______________________ ________________________

Divide. Write each answer in simplest form. The first one is done for you.

7. 7 2110 6

8. 52 17

_____________________________________ _____________________________________

9. 3 24 25 5

10. 11 3112 4

_____________________________________ _____________________________________

Write a division expression for each problem. Then solve. The first one is done for you.

11. Larry has 395

yards of fabric. He will use 225

yards to make each vest.

How many vests can Larry make?

____________________________________________ Larry can make ________ vests.

12. A patio has an area of 5206

ft2, and the width is 3 12

feet. What is the

length of the patio?

____________________________________________________ The patio is ________ feet long.

LESSON

4-3

( )9 2 1 192 2

+

The reciprocal is 219

.

7 8 7 6 42 2110 6 10 8 80 40

3 2 48 12 48 5 489 2 45 5 5 5 5 12 12

4

Name ________________________________________ Date __________________ Class __________________


67

Dividing Mixed Numbers Reteach

Two numbers are reciprocals if their product is 1.

73

and 37

are reciprocals because 7 3 1.3 7

Write a mixed number as an improper fraction to find its reciprocal.

324

and 411

are reciprocals because 3 1124 4

and 11 4 14 11

.

To find 3 32 14 4

, first rewrite the mixed numbers as improper fractions.

11 74 4

Next, rewrite the expression as a multiplication expression and replace the divisor with its reciprocal.

11 44 7

Solve. Write your answer in simplest form.

3 3 11 7 11 4 11 42 1 14 4 4 4 4 7 7 7

Find the reciprocal.

1. 914

2. 132

3. 2103

________________________ _______________________ ________________________

Complete the division. Write each answer in simplest form.

4. 3 13 25 4

5. 1 11 12 4

6. 5 7112 8

18 5 4

5 9

3 2 4

12 8

________________________ ________________________ ________________________

7. 1 138 2

8. 1 21 26 3

9. 12 15

________________________ ________________________ ________________________

LESSON

4-3

Name ________________________________________ Date __________________ Class __________________


68

Dividing Mixed Numbers Reading Strategies: Use a Model

A model is useful for dividing mixed numbers.

The Smith family has a 2 12

-foot-long sandwich to share. Each

12

-foot of the sandwich serves one person. How many 12

-foot

servings are in this sandwich?

Find 2 1 12 2

.

Step 1: Draw a square and label it 12

.

12

Step 2: Draw a row of these squares until they add up to 2 12

.

12

12

12

12

12

Step 3: Count the number of squares needed to reach 2 12

.

12

12

12

12

12

1. How do you represent a single serving?

_________________________________________________________________________________________

2. Why draw a row of servings until they add up to 2 12

?

_________________________________________________________________________________________

3. How many 12

-foot servings does the Smith family have?

_________________________________________________________________________________________

4. What is 2 1 12 2

? ________________

LESSON

4-3

2 12

2 12

1 2 3 4 5

Name ________________________________________ Date __________________ Class __________________


69

W ?

Dividing Mixed Numbers Success for English Learners

Problem 1

What is the reciprocal of 324

?

Write the mixed number as an improper fraction.

3 32 24 4

8 3 4 411 4

+

+

What is the reciprocal of 114

?

Flip it!

114

411

How can I tell this is right?

11 4 44 14 11 44

Problem 2 How wide is the rectangle?

18 ft2

l

What is the area?

What is the length?

How can I find the width?

Divide the area by the length.

2 1 170 1756 83 2 3 2

170 2 3 17

10170 23 171

20 2 or 63 3

The width is 263

ft.

1. How is dividing mixed numbers different from multiplying mixed numbers?

_________________________________________________________________________________________

_________________________________________________________________________________________

2. What is the first step to divide mixed numbers?

_________________________________________________________________________________________

3. Why would you expect the width of the rectangle to be about 7 ft?

_________________________________________________________________________________________

_________________________________________________________________________________________

2256 ft3

A

LESSON

4-3

2256 ft3

18 ft2

Wow! The product is 1.

Name ________________________________________ Date __________________ Class __________________


70

Solving Multistep Problems with Fractions and Mixed Numbers Practice and Problem Solving: A/B

Solve. Show your work.

1. After a holiday dinner, there are 133

apple pies left and 526

pumpkin

pies left. a. How much more apple pie than pumpkin pie is left?

_____________________________________________________________________________________

b. Tom ate 12

of the leftovers. How much pie in all did he eat?

_____________________________________________________________________________________

2. An angelfish was 112

inches long when it was bought. Now it is

123

inches long.

a. How much has the angelfish grown? _________________________________________________

b. An inch is 112

of a foot. How much has the angelfish grown in feet? ____________________

3. There was a 6 square-foot piece of wrapping paper for a birthday present.

It takes 338

square feet of the paper to wrap the present. How many

pieces of 6 square-foot paper are needed to wrap 3 of these presents?

_________________________________________________________________________________________

4. Today, a bicycle rider rode her bike 152

miles. Yesterday, she rode

164

miles. The difference in length between the two rides is what

fraction of the longer ride?

_________________________________________________________________________________________

5. A survey by the state health department found that the average person

ate 208 pounds of vegetables last year and 51258

pounds of fruit. What

fraction of the total pounds of fruit and vegetables do the pounds of fruits represent?

_________________________________________________________________________________________

LESSON

4-4

Name ________________________________________ Date __________________ Class __________________


71

Solving Multistep Problems with Fractions and Mixed Numbers Practice and Problem Solving: C

Solve. Show your work.

1. One wintry week a ski town got 1152

inches of snow. On Monday the town

got 324

inches; on Tuesday it got 112

times as much; and on Wednesday it

got 78

inch. How much snow did the town get the rest of the week?

_________________________________________________________________________________________

2. Laura was making a recipe that said the ingredients were for 6 people,

but she needed to make it for 8 people. The recipe called for 223

cups

of milk and 14

cup of oil. How many cups of these liquid ingredients did

she need for 8 people?

_________________________________________________________________________________________

3. Brian decided to make the same recipe in Exercise 2 for 4 people. How many cups of these liquid ingredients did he need? Explain two ways you could find the answer.

_________________________________________________________________________________________

_________________________________________________________________________________________

4. A fence around a square garden is made up of 4 equal-sized pieces

that are each 152

feet long. Matt decided to make the sides 122

times

longer. How much fencing does he need in all?

_________________________________________________________________________________________

5. Pedro spent 125

hours on his math homework and half as long on his

science homework one weekend. If his English and social studies

homework took 78

as long, how much time did he spend on homework

that weekend?

_________________________________________________________________________________________

6. Ken spent 15

of his allowance on a movie, 38

on snacks, and 27

on

games. If his allowance was $20, how much did Ken have left?

_________________________________________________________________________________________

LESSON

4-4

Name ________________________________________ Date __________________ Class __________________


72

Solving Multistep Problems with Fractions and Mixed Numbers Practice and Problem Solving: D

Solve. Show your work. The first one is done for you.

1. After a party, there is 13

apple pie left and 56

pumpkin pie left.

a. How much more pumpkin pie than apple pie is left?

_____________________________________________________________________________________

b. Terri ate 13

of the leftover pies. How much pie did she eat?

_____________________________________________________________________________________

2. An angelfish was 112

inches long when it was bought. Now it is

122

inches long.

a. How much has the angelfish grown? _________________________________________________

b. An inch is 112

of a foot. How much has the angelfish grown in feet? ____________________

3. There was a 6 square-foot piece of wrapping paper for a present. It

takes 138

square feet of paper to wrap the present. How many pieces

of 6-foot-square paper are needed to wrap 2 presents?

_________________________________________________________________________________________

4. Today, a bicycle rider rode her bike 152

miles. Yesterday, she rode

6 miles. What fraction of the total of the two rides is the longer ride?

_________________________________________________________________________________________

5. A survey by the state health department found that the average person

ate 208 pounds of vegetables last year and 51258

pounds of fruit. How

many pounds of vegetables and fruits did the average person eat per

month? [Hint: Remember that 1 month is 112

of a year.]

_________________________________________________________________________________________

LESSON

4-4

5 1 5 2 3 1 6 3 6 6 6 2

or pie− −

Name ________________________________________ Date __________________ Class __________________


73

Solving Multistep Problems with Fractions and Mixed Numbers Reteach

In order to solve some problems involving mixed numbers, you will have to rewrite the mixed number as a whole number and an improper

fraction. For example, 123

can be rewritten as 413

. The two numbers

are the same because 123

1 + 113

33

+ 113

1 + 33

+ 13

.

This step is necessary when subtracting mixed numbers as shown here.

Example

After an office party, 143

pizzas are left. A day later, there are 516

pizzas left. How much pizza

was eaten during the day after the party? One third of the pizza eaten on the day after the party was pepperoni. How much of the day-old pizza eaten was pepperoni?

Solution: First, change the denominator to the common denominator of 6:

143

246

Then, write the subtraction problem: 246

−516

Since the fraction with 4 is less than the fraction with 1, write 246

as 1 + 236

and write 1 as 66

so that the subtraction problem

becomes 836

−516

.

Subtract the whole numbers and subtract the numerators of the fractions:

836

−516

2 + 36

or 122

; 122

pizzas were eaten during the next day. Of

these, 13

were pepperoni. So: 1 1 5 1 52 .2 3 2 3 6

On the second day, 56

of a pepperoni pizza was eaten.

Solve by rewriting the mixed number that is being subtracted.

1. A deli ordered 162

wheels of cheese. Over the weekend, 538

wheels of

cheese were sold. On Tuesday another 314

wheels were sold. How

much cheese was left for Wednesday?

_________________________________________________________________________________________

LESSON

4-4

Name ________________________________________ Date __________________ Class __________________


74

Solving Multistep Problems with Fractions and Mixed Numbers Reading Strategies: Compare and Contrast

When you solve problems with fractions, mixed numbers, and whole numbers, it can be helpful to compare and contrast different methods of solving the problems. The example shows two ways to solve a problem involving a mixed number and a whole number.

Example A carpet store puts 40 square yards of outdoor carpet on sale. After the

sale, they have 1103

square yards of carpet left. How much did they sell?

Of the carpet sold, half was green. How many square feet of green carpet did they sell?

Method 1 Estimation can be used to solve this problem without converting the 40 yards to a fraction and a mixed number. Forty yards less 10 yards is 30 yards. However, the real difference is one third of a yard less than 30 yards. Thirty yards less one third of a yard is 29 yards and a fraction. What fraction? One third from one whole leaves two thirds. So, the amount of

carpet sold is 29 yards and two thirds of a yard or 2293

square yards. Half

of 2293

square yards is about 15 square yards or 1142

square yards plus

13

square yard. So, they sold 5146

square yards of green carpet.

Method 2 Change 40 yards to 39 yards and one yard. Write one yard as

thirds: three thirds. So, the problem becomes 39 + 33

minus 1103

. Subtract

the whole numbers: 39 − 10 29. Subtract the fractions. Since the fractions have the same denominators, the numerators can be subtracted:

33

− 13

23

.

Add 29 and 23

to get 29 23

yards.

Then multiply 2293

by 12

to get 5146

square yards.

Answer the question. 1. Which of the two methods of doing the problem in the example do you

prefer and why?

_________________________________________________________________________________________

_________________________________________________________________________________________

LESSON

4-4

Name ________________________________________ Date __________________ Class __________________


75

Solving Multistep Problems with Fractions and Mixed Numbers Success for English Learners

Problem 1a Sarai has a piece of ribbon that is 16

3yards long. She cuts off a piece that

is 526

yards long. How much ribbon is left?

Rewrite the fraction:

−1 56 23 6

−2 5 3 16 2 3 36 6 6 2

or yards

Problem 1b Sarai wants to use 1

2of a 13

2-yard piece of ribbon to wrap a present.

How much ribbon will she use?

1 1 7 132 2 2 2

7 or431 yards4

1. Why is 6 the common denominator in Problem 1a?

_________________________________________________________________________________________

2. What operations did you use to solve Problems 1a and 1b?

_________________________________________________________________________________________

LESSON

4-4

Step 1 Subtract to find out how much is left.

Name ________________________________________ Date __________________ Class __________________


76

Operations with Fractions Challenge

The table shows the length and width of 4 rug designs that a carpet store stocks. Use the table to answer problems 1–2.

Rug Design Length (ft) Width (ft)

Classic 182

3104

Deco 3104

398

Solid 275

385

Modern 3105

192

1. The price of each rug is found by multiplying the area of the rug (length times width) by the price per square foot. The price for all 4 rug designs listed above is $8 per square foot. Which rug is the most expensive? How much does it cost?

_________________________________________________________________________________________

2. Pauline orders a custom rug. She wants a rug that is the same final price as the Deco but the same width as the Modern. What is the length of the rug Pauline wants to purchase? Explain.

_________________________________________________________________________________________

_________________________________________________________________________________________

Solve.

3. 12

, 23

, 34

, 45

, 56

, … 99100

In the list above, each fraction after the first is obtained by adding 1 to both the numerator and denominator of the fraction before it.

For example, the first fraction is 12

. To get the second fraction, add

1 to 1 and to 2: 1 12 1

++

23

. This pattern continues to 99100

. What is

the product of the fractions in the list above? What pattern can help you find the product quickly?

_________________________________________________________________________________________

_________________________________________________________________________________________

MODULE

4

Name ________________________________________ Date __________________ Class __________________


77

Dividing Whole Numbers Practice and Problem Solving: A/B

Estimate each quotient by rounding the dividend and the divisor to the largest place value. 1. 585 13 2. 2,756 53 3. 22,528 98

________________________ _______________________ ________________________

4. 7,790 210 5. 17,658 360 6. 916 320

________________________ _______________________ ________________________

Find each quotient using long division. Show your work.

7. 8. 9.

________________________ _______________________ ________________________

Find each quotient and remainder using long division. Show your work.

10. 11. 12.

________________________ _______________________ ________________________

Solve. 13. At the museum, there were 4,050 students in attendance from a total

of 15 different school districts. What was the average attendance from each school district?

_________________________________________________________________________________________

14. The Appalachian Trail is about 2,175 miles long. If a hiker averages 12 miles each day, how long will it take her to hike the length of the trail?

_________________________________________________________________________________________

LESSON

5-1

______29 1,334

______92 20,884

______25 18,175

______18 2,902

______64 34,680

______215 52,245

Name ________________________________________ Date __________________ Class __________________


78

Dividing Whole Numbers Practice and Problem Solving: C

Estimate each quotient by rounding the dividend and the divisor to the largest place value. Then estimate each quotient by rounding the dividend and divisor to the nearest 10. 1. 585 13 2. 2,756 53 3. 22,528 98

__________________ __________________ __________________

__________________ __________________ __________________

4. 7,790 210 5. 17,658 360 6. 916 320

__________________ __________________ __________________

__________________ __________________ __________________

Answer the question. 7. In Exercises 1−6, which method of estimating usually gave you an

estimated quotient closer to the quotient?

_________________________________________________________________________________________

Each quotient is given with the remainder in decimal form. What is the remainder for each in fraction form?

8. 9. 10.

__________________ __________________ __________________

Solve. 11. One weekend, 5,780 people saw a new movie at 17 different theaters. Each

theater sold tickets at $7.50 a piece. Assuming that each theater received the same number of moviegoers, how much did each theater make?

_________________________________________________________________________________________

12. The Appalachian Trail is about 2,175 miles long. a. If Katia averages 18 miles each day, how long will it take her to

hike the length of the trail?

__________________

b. Katia’s friend Joelle joins her for the last quarter portion of the hike. How many full days will Joelle hike?

__________________

LESSON

5-1

541.87564 34,680

243.065116215 52,259

161.218 2,902

Name ________________________________________ Date __________________ Class __________________


79

Dividing Whole Numbers Practice and Problem Solving: D

Estimate the quotients by filling in the blanks. The first one is done for you. 1. 345 28 2. 1,711 105

345 rounds to ___________ hundreds. 1,711 rounds to ____________ thousands.

28 rounds to ____________ tens. 105 rounds to _____________ hundred.

Quotient estimate: Quotient estimate:

________________________________________ ________________________________________

3. 715 24 4. 2,315 ÷ 95

________________________________________ ________________________________________

Use long division to find the quotients and remainders by writing numbers in the boxes. The first one is done for you.

5. 6.

7. 8.

LESSON

5-1

300 30, or about 10

3

2 3 7 6 4

4

5

4 1 1 8 7 5

1 5 3 2 5

5

2 1

3 0

2

1 5

1 0

4

1

3 6 8 4 4 1

3

Name ________________________________________ Date __________________ Class __________________


80

Dividing Whole Numbers Reteach

Division is used to separate a quantity into a given number of equal parts. It is also used to separate a quantity into parts of a specific size.

A division algorithm breaks division with greater numbers into a series of lesser divisions. Follow the steps for each lesser division:

Step 1: Divide and write the number in the first correct place in the quotient. Step 2: Multiply the divisor by the number in the quotient. Step 3: Subtract. Step 4: Bring down the next digit in the dividend. Repeat these steps until there are no digits from the dividend left to bring down.

Jon bought a package of 792 labels. There are 24 sheets of labels in the package. How many labels are on each sheet?

Divide to find the number of labels per sheet. 792 labels 24 sheets

Problem 1 Divide. 79 24 3. Place 3 in the tens place. Multiply. 24 3 72 Subtract. 79 − 72 7 Bring down the next digit in the dividend: 2. Problem 2 Repeat the process. Divide.72 24 3. Place 3 in the ___________ place. Multiply. 24 3 72 Subtract. 72 − 72 0

792 24 33. There are 33 labels on each sheet.

Use the 4-step process to do the division. 1. The art teacher has a box of 473 markers. She wants to distribute them

evenly among 11 tables. How many markers will she put on each table?

Divide: 47 ________ ________ Multiply: 11 ________ ________ Subtract: 47 − ________ ________ Bring down the ________. Repeat the steps. Divide: ________ ________ ________ Multiply: ________ ________ ________ Subtract: ________ − ________ ________ Answer: ________ markers

24 792

3324 792

72

7272

0

−

− ones

−

repeat

−

repeat

11 47344

3333

0

−

−

LESSON

5-1

−

repeat

Name ________________________________________ Date __________________ Class __________________


81

Dividing Whole Numbers Reading Strategies: Build Vocabulary

Division is the act of separating something into equal parts. This means that the parts of a problem include a total, the number of groups, and the number of items in one group. These three parts are named the dividend, divisor, and quotient.

The number that represents the total or what is being separated, is the dividend.

Identify the number that is the dividend for each situation. 1. Gina has a collection of 645 stamps. She stores the stamps on pages

that each holds 15 stamps.

_________________________________________________________________________________________

2. Karl wants to fill a storage case that he uses to hold his miniature toy cars. The case has 5 racks that will hold 120 cars.

_________________________________________________________________________________________

Unlike the dividend, the divisor and quotient can take on multiple roles. They can each represent the number of groups or the number of items in the group. When the divisor is the number of groups, the quotient is the number of items in each group. When the divisor is the size of the group, the quotient is the number of groups. These differences do not change the way you compute the quotient, it simply changes the meaning of the quotient.

Answer each question. Explain your answer. 3. In Question 1, does the number 15 represent the number of groups or

the number of items in a group?

_________________________________________________________________________________________

4. In Question 2, does the number 5 represent the number of groups or the number of items in a group?

_________________________________________________________________________________________

Find each quotient. Does the quotient represent the number of groups or the number of items in a group? 5. Gina has a collection of 645 stamps. She stores the stamps on pages

that each holds 15 stamps. How many pages are in Gina’s collection?

_________________________________________________________________________________________

6. Karl uses a storage case to hold his miniature toy cars. The case has 5 racks that will hold 120 cars. How many cars will each of the 5 racks hold?

_________________________________________________________________________________________

LESSON

5-1

Name ________________________________________ Date __________________ Class __________________


82

Dividing Whole Numbers Success for English Learners

Problem 1 Step 1 Estimate 120 65 120 120 65 60

Step 2 120 65 120 60 120 60 2

So, the quotient is about 2.

Check your estimate. 120 65 1.846 So, 1.846 is close to 2.

Problem 2 Divide 235 by 17.

1317 235

176551

1 4

−

−

Step 1 Divide 23 by 17. What do you get? 1 with 6 left over. Write the 1 in the quotient. Multiply 1 times 17. Write 17 below the 23. Subtract 17 from 23. Step 2 Write the 6 and bring down the 5. What do you get? 65. Step 3 Divide 65 by 17. What do you get? 3 with 14 left over. Write the 3 in the quotient. Step 4 Write the answer: 235 17 13 R 14

1. In Problem 1, is the estimate an underestimate or an overestimate?

_________________________________________________________________________________________

2. Why are 150 and 50 compatible numbers?

_________________________________________________________________________________________

3. In Problem 2, write the answer in Step 4 with a whole number and a fraction.

_________________________________________________________________________________________

LESSON

5-1

Think:

12 6 2 and 120 60 2.

So, 120 and 60 are compatible numbers.

They are easy to divide mentally.

Name ________________________________________ Date __________________ Class __________________


83

Adding and Subtracting Decimals Practice and Problem Solving: A/B

Find each sum or difference. 1. 1.5 + 2.3 2. 8.9 − 5.1 3. 2.5 + 1.3 + 4.1

__________________ __________________ __________________

4. 7.25 + 8.75 5. 8.16 − 7.72 6. 3.3 + 4.5 + 2.6

__________________ __________________ __________________

7. 8.9 + 3.05 8. 10.64 − 8.8 9. 4.1 + 0.35 + 6.564

__________________ __________________ __________________

Solve. 10. Marcus is 1.5 meters tall. His sister is 0.1 meter taller than Marcus.

Their father is 0.2 meter taller than his sister. How tall is their father?

_________________________________________________________________________________________

11. Jennifer brought $24.75 to the baseball game. She spent $12.45 on drinks and snacks. How much money does she have left over?

_________________________________________________________________________________________

Find the missing digit. 12. 13. 14.

_________ _________ _________

Solve. 15. A gourmet pizza café sells three sizes of pizzas. If you buy all three

sizes, it costs $46.24. A medium pizza costs $15.75 and a large pizza costs $17.50. How much does the small pizza cost?

_________________________________________________________________________________________

16. A carpenter has three sheets of plywood that are each 6.85 feet long. A 3.4-foot piece is cut from one sheet and 0.5-foot piece is cut from another sheet. How many feet of plywood is left in all?

_________________________________________________________________________________________

LESSON

5-2

7.0892. 139.502

+

16.594.175

11.419−

6.2 679.75

15.9867+

Name ________________________________________ Date __________________ Class __________________


84

Adding and Subtracting Decimals Practice and Problem Solving: C

Write each fraction as a decimal.

1. 23

2. 76

3. 19

3 4. 518

5

____________ ____________ ____________ ____________

5. What is a common feature of all of the decimals in Exercises 1–4?

_________________________________________________________________________________________

Add or subtract as indicated. 6. 0.333 7. 0.121212 8. 3.456456 + 0.666 + 0.454545 − 1.345345

____________ ____________ ____________

Solve. 9. Write 0.333 as a fraction. 10. Write 0.666 as a fraction.

n _________ n _________

10n _________ 10n _________

9n _________; n _________ 9n _________; n _________

11. Write 0.343434 as a fraction. 12. Write 1.432432 as a fraction.

n _________ n _________

100n _________ 1,000n _________

99n _________; n _________ 999n _________; n _________

13. Why are the decimals in Exercises 11 and 12 multiplied by 100 and 1,000, respectively?

_________________________________________________________________________________________

LESSON

5-2

Name ________________________________________ Date __________________ Class __________________


85

Adding and Subtracting Decimals Practice and Problem Solving: D

Shade the grid to find each sum. The first one is done for you. 1. 0.3 + 0.34 2. 0.15 + 0.19

__________________ __________________

Estimate the answer. The first one is done for you.

3. 12.05 is about ______ 4. 34.5 is about ______

+ 27.6 is about ______ − 17.7 is about ______

______ or about ______ ______ or about ______

Fill in the digits and find the answer. The first one is done for you. 5. 47.65 + 8.059 6. 77.5 + 23.87

____________________________ ____________________________

Find the sum or difference. Use any of the methods in this lesson. The first one is done for you. 7. 105.6 − 78.25 8. 13.2 + 0.5 + 200.6

105.60

− 78.25

27.35

____________________________ ____________________________

LESSON

5-2

0.64

28

12

40

55.709

39.65

27.35

Name ________________________________________ Date __________________ Class __________________


86

Adding and Subtracting Decimals Reteach

You can use a place-value chart to help you add and subtract decimals.

Add 1.4 and 0.9. Subtract 2.4 from 3.1.

So, 1.4 + 0.9 2.3. So, 3.1 − 2.4 0.7.

Find each sum or difference. 1. 2.

_________________ __________________

3. 4.3 + 1.4 4. 14.4 − 3.8 5. 7.3 + 8.5 6. 12.34 − 6.9

________________ ________________ _______________ ________________

Estimate the answers to Exercises 3–6 by rounding to the nearest whole number. Compare your estimate to the exact answers. 7. 4.3 + 1.4 8. 14.4 − 3.8 9. 7.3 + 8.5 10. 12.34 − 6.9

________________ ________________ _______________ ________________

________________ ________________ _______________ ________________

LESSON

5-2

Name ________________________________________ Date __________________ Class __________________


87

Adding and Subtracting Decimals Reading Strategies: Use a Graphic Organizer

Writing decimals in a place-value grid helps you line up decimal points to add or subtract decimals.

1. How does the place-value grid help you add or subtract?

_________________________________________________________________________________________

2. Place these numbers on the place-value 3. Place this problem on the place-value grid below: 3.25, 1.06, 2.9. grid below: 23.8−7.2.

4. Add the numbers on the place-value 5. Subtract the numbers on the place- grid in Exercise 2. What is the sum? value grid in Exercise 3. What is the difference?

________________________________________ ________________________________________

6. For which numbers did you add zero as a place holder?

_________________________________________________________________________________________

LESSON

5-2

Add zeros as place holders.

Place decimal point in answer.

1.405.382.709.48

+

28.056.30

21.75−

Name ________________________________________ Date __________________ Class __________________


88

Adding and Subtracting Decimals Success for English Learners

Problem 1 Place value charts can help you add or subtract decimals.

Example

Add 1.05 and 0.0517.

Solution

Look at the place value charts for the numbers.

When you add, you line up the decimal points: 1.05

+ 0.0517

1.1017

Problem 2 Add 0.12 and 0.50.

First, line up the decimal points:

Second, add the numbers in each place. 0, 6, and 2

Third, write the answer: 0.62

1. In Problem 1, how would read the decimal point if you are saying the number words?

_________________________________________________________________________________________

2. What happens to the value of the sum or difference of numbers if you do not align their decimal points?

_________________________________________________________________________________________

_________________________________________________________________________________________

LESSON

5-2

Name ________________________________________ Date __________________ Class __________________


89

Multiplying Decimals Practice and Problem Solving: A/B

Show the decimal multiplication on the grids. Find the product. 1. 0.2 0.6× ________ 2. 0.3 × 0.7 ________

Draw an area model to represent the multiplication problems below. Find the product. 3. 1.2 × 3.3 = ______ 4. 4.1 × 2.1 = ______

Multiply. 5. 0.1 6. 0.9 7. 0.3 8. 1.6 × 0.2 × 6 × 0.8 × 2.9

9. 1.5 × 0.41 = 10. 0.24 × 2.68 = 11. 3.13 × 4.69 = 12. 5.48 × 15.12 =

_________________ _________________ _________________ ________________

Solve. 13. Each basket can hold 2.5 pounds of apples. How many pounds can

7 baskets hold?

_________________________________________________________________________________________

14. Canvas cloth costs $7.50 per square meter. How much will 3.5 square meters of canvas cost?

_________________________________________________________________________________________

LESSON

5-3

Name ________________________________________ Date __________________ Class __________________


90

Multiplying Decimals Practice and Problem Solving: C

Estimate each product to the nearest whole number. Then, find the product. 1. 0.7 0.85 2. 3.05 1.95 3. 0.55 2.3

________________________ _______________________ ________________________

4. 4.699 1.74 5. 10.37 5.086 6. 5.593 19.71

________________________ _______________________ ________________________

Compare using or without calculating the product.

7. 2.4 3.8 3.5 2.8 8. 6.28 3.82 3.3 6.84

Solve. 9. A forestry service biologist has time to study insect infestation in an

area of 50 square kilometers. On the forest service map, the scale is 1 centimeter equals 1 kilometer. The four possible sectors available for study appear as rectangles on the map. Complete the table by calculating the map area of each sector.

Sector Map Dimensions (cm) Map Area (cm2)

A 2.5 5.8

B 3.7 2.1

C 4.7 3.5

D 4.2 2.8

a. How can you calculate the actual area of each sector?

_____________________________________________________________________________________

b. Does the biologist have time to study all four areas? Explain why or why not?

_____________________________________________________________________________________

_____________________________________________________________________________________

c. What combinations of three sectors could the scientist study?

_____________________________________________________________________________________

d. Which combination of sectors maximizes the area to be studied? Explain why.

_____________________________________________________________________________________

LESSON

5-3

Name ________________________________________ Date __________________ Class __________________


91

7.1 52.5×

4.361.2×

Multiplying Decimals Practice and Problem Solving: D

Multiply. The first one is done for you.

1.

0.5

3×

2.

40.8×

3.

9

0.7×

4.

0.253×

.1 5

Show the decimal multiplication on the grids. Do not solve. 5. 0.1 0.7× 6. 0.4 0.8× 7. 0.3 0.7×

Name the number of decimal places. 8. 9.

Multiply. The first one is done for you. 10. 11.

3575 1430 17.875

Solve. 12. A cabinetmaker buys 3.5 liters of oak varnish. The varnish costs

$4.95 per liter.

a. Write a multiplication expression for this purchase.

_____________________________________

b. How much does 3 liters of varnish cost? _________________________

c. How much does 0.5 liters of varnish cost? ___________________________

d. What is the total cost of 3.5 liters of varnish? ___________________________

LESSON

5-3

2.4 _____ decimal places0.83 _____ decimal places

Answer _____ total decimal places

←× ←

←

0.456 _____ decimal places2.4 _____ decimal places

Answer _____ total decimal places

←× ←

←

Name ________________________________________ Date __________________ Class __________________


92

Multiplying Decimals Reteach

You can use a model to help you multiply a decimal by a whole number.

Find the product of 0.12 and 4.

Use a 10-by-10 grid. Shade 4 groups of 12 squares.

Count the number of shaded squares. Since you have shaded 48 of the 100 squares, 0.12 4 0.48.

Find each product. 1. 0.23 3 2. 0.41 2 3. 0.01 5 4. 0.32 2

_________________ _________________ _________________ ________________

5. 0.15 3 6. 0.42 2 7. 0.04 8 8. 0.22 4

_________________ _________________ _________________ ________________

You can also use a model to help you multiply a decimal by a decimal.

Find the product of 0.8 and 0.4.

Step 1 Shade 8 tenths of the figure.

Step 2 Shade darker 4 tenths of the shaded area.

Step 3 How many squares have you shaded twice?

You have twice shaded 32 of the squares.

So, 0.8 0.4 0.32.

Find each product. 9. 0.2 0.8 10. 0.7 0.9 11. 0.5 0.5 12. 0.3 0.6

_________________ _________________ _________________ ________________

13. 0.5 0.2 14. 0.4 0.4 15. 0.1 0.9 16. 0.4 0.7

_________________ _________________ _________________ ________________

LESSON

5-3

Name ________________________________________ Date __________________ Class __________________


93

Multiplying Decimals Reading Strategies: Use Graphic Aids

Each grid has 26 of 100 squares shaded to represent 0.26.

You can add the decimals to find out how much of the grids are shaded.

Or, you can multiply the shading of one grid by 6.

0.26 + 0.26 + 0.26 + 0.26 + 0.26 + 0.26 1.56

0.26 6 1.56

Use the grids for Exercises 1 to 5.

1. Shade the grids so that each one represents 0.89.

2. Write an addition expression to represent the shaded grids.

_________________________________________________________________________________________

3. Evaluate your addition expression to find the sum.

_________________________________________________________________________________________

4. Write a multiplication expression to represent the shaded grids.

_________________________________________________________________________________________

5. Evaluate your multiplication expression to find the product.

_________________________________________________________________________________________

LESSON

5-3

Name ________________________________________ Date __________________ Class __________________


94

Multiplying Decimals Success for English Learners

Problem 1 How many decimal places are in each number?

2.7 1.25 23

Start at the decimal point. Count the digits to its right.

one decimal place two decimal places no decimal place

Problem 2 Multiply: 1.2 × 1.6

1.2 1 decimal place 1.2

1.6 1 decimal place × 1.6

72

+ 120

1 + 1 = 2 decimal places 1.92

1. How do you find the decimal place in the product of two decimals?

_________________________________________________________________________________________

2. To place the decimal point in the product of two decimals, do you move the decimal point to the left or to the right?

_________________________________________________________________________________________

3. After you place the decimal point in a product of two decimals, how do you tell if the answer is reasonable?

_________________________________________________________________________________________

Is each product reasonable? Write yes or no. If no, give a reasonable estimate.

4. 0.8 × 3 = 0.12 5. 5.2 × 6.7 = 34.84 6. 2.4 × 3 = 72

_________________ _________________ _________________

LESSON

5-3

2.7

1.25

23

Name ________________________________________ Date __________________ Class __________________


95

Dividing Decimals Practice and Problem Solving: A/B

Use decimal grids to find each quotient. First, shade the grid. Then, separate the model to show the correct number of equal parts.

1. 3.6 1.2 2. 3.27 3

________________________________________ ______________________________________

Find each quotient.

3. 9.5 142.5 4. 3 39.6 5. 2 10.88

________________________ _______________________ _________________

6. 10.5 1.5 7. 9.75 1.3 8. 37.5 2.5

________________________ _______________________ _________________

Estimate each quotient to the nearest whole number. Then, find the actual quotient. 9. 0.9 3.78 10. 2.5 36 11. 0.25 7 12. 9.5 142.5

_________________ _________________ _________________ ________________

Solve. 13. A camera attached to a telescope photographs a star’s image once

every 0.045 seconds. How many complete images can the camera capture in 3 seconds?

_________________________________________________________________________________________

14. A geologist noticed that land along a fault line moved 24.8 centimeters over the past 175 years. On average, how much did the land move each year?

_________________________________________________________________________________________

LESSON

5-4

Name ________________________________________ Date __________________ Class __________________


96

Dividing Decimals Practice and Problem Solving: C

Estimate the quotient. Then find the exact quotient. 1. 8.4 2.4 2. 13.75 2.25 3. 5.45 0.5

Estimate: Estimate: Estimate:

_________________ _________________ _________________

Exact Quotient: Exact Quotient: Exact Quotient:

_________________ _________________ _________________

Compare using , , or without calculating the quotient.

4. 0.05 3 { 0.005 3 5. 1.9 4.7 { 19 4.7

6. 0.35 0.78 { 0.35 7.8 7. 1.2 34 { 0.12 3.4

Solve. 8. Acme Hardware is introducing a new product called Greener Cleaner.

Complete the table by finding the cost per milliliter for each size based on the sales price. One liter is 1,000 milliliters.

Size Amount of Liquid Sale Price Price

per Milliliter

Small 250 milliliters $4.50

Medium 500 milliliters $9.95

Large 1 liter $16.95

a. Write an expression using or to compare the three containers by price per milliliter.

_____________________________________________________________________________________

b. What is the least expensive way to buy 1,500 milliliters of Green Cleaner? Write an expression to represent your choice and evaluate.

_____________________________________________________________________________________

c. What is the most expensive way to buy 1,500 milliliters of Green Cleaner? Write an expression to represent your choice and evaluate.

_____________________________________________________________________________________

LESSON

5-4

Name ________________________________________ Date __________________ Class __________________


97

Dividing Decimals Practice and Problem Solving: D

Find each quotient. The first one is done for you. 1. 2.8 4 2. 1.8 2 3. 3.6 6

_________________ _________________ _________________

4. 7.2 9 5. 0.15 3 6. 4.8 8

_________________ _________________ _________________

Find each quotient. The first one is done for you. 7. 2.4 0.4 8. 1.4 0.2 9. 4.8 0.6

_________________ _________________ _________________

10. 3.3 0.3 11. 2.6 1.3 12. 7.2 1.2

_________________ _________________ _________________

Solve. 13. At the grocery store, a six-pack of bottled water costs $2.88.

How much does each bottle cost?

_________________________________________________________________________________________

14. It rained 2.79 inches in July. What was the average daily rainfall in July? (Hint: July has 31 days.)

_________________________________________________________________________________________

15. Over several months, a meteorologist recorded a total snowfall of 8.6 centimeters. During this period, the average monthly snowfall was 4.3 centimeters. For how many months did the meteorologist collect measurements of the snowfalls?

_________________________________________________________________________________________

16. Almonds cost $3.49 per pound. A bag of almonds costs $6.95. To the nearest whole pound, about how many pounds of almonds are in the bag?

_________________________________________________________________________________________

0.7

6

LESSON

5-4

Name ________________________________________ Date __________________ Class __________________


98

Dividing Decimals Reteach

You can use decimal grids to help you divide by whole numbers.

To divide 0.35 by 7, first shade in a decimal grid to show thirty-five hundredths.

0.35 7 means “divide 0.35 into 7 equal groups.” Show this on the decimal grid.

The number of units in each group is the quotient.

So, 0.35 7 0.05.

Find each quotient. 1. 0.6 5 2. 0.78 6 3. 0.32 4 4. 0.99 0.0033

________________ ________________ _______________ ________________

You can use powers of 10 to help you divide a decimal by a decimal.

Divide 0.048 by 0.12.

Notice that 0.12 has two decimal places. To make this a whole number, multiply by 100.

0.048 0.12 0.12 100 12 0.048 100 4.8

Then divide.

4.8 120.4

12 4.84 8

0

Step 1: Divide as you would with a whole number. Step 2: Think 48 12 4. Step 3: Place the decimal point in the quotient. Add a zero as necessary.

So, 0.048 0.12 0.4.

Find each quotient. 5. 0.4 0.08 6. 0.9 0.63 7. 0.008 0.4 8. 0.04 0.032

________________ _________________ _________________ _________________

LESSON

5-4

Name ________________________________________ Date __________________ Class __________________


99

Dividing Decimals Reading Strategies: Use Graphic Aids

You can use a hundreds grid to show division with decimals.

The grid shows 0.15.

0.15 3 means “separate 0.15 into 3 equal groups.”

0.15 3 makes 3 equal groups of 0.05.

0.15 3 0.05

Use the grids to complete 1–8. 1. Shade 0.60 of the grid at right.

2. Divide the shaded area into 3 equal sections.

3. Write a decimal that represents each section. ____________

4. Write a division problem for your model.

–––––––––––––––––––––––––––––––––––––––––––

5. Shade 0.72 of the grid at right.

6. Divide the shaded area into 8 equal sections.

7. Write a decimal that represents each section. ___________

8. Write a division problem for your model.

–––––––––––––––––––––––––––––––––––––––––––

LESSON

5-4

Name ________________________________________ Date __________________ Class __________________


100

Dividing Decimals Success for English Learners

Problem 1 Find 3.6 1.2.

1 . 2 3 . 6 12 36

Divisor Dividend

Remember: To multiply by 10, move the decimal point 1 place to the right.

Problem 2 Sari’s car goes 17.5 miles for every gallon of gas. How many gallons of gas does Sari’s car use to go 227.5 miles?

13.0

17.5 227.5

1. What is the quotient in Problem 1?

_________________________________________________________________________________________

2. Does the quotient in Problem 1 have a remainder? How do you know?

_________________________________________________________________________________________

3. Write another question to go with Problem 2. Solve.

_________________________________________________________________________________________

_________________________________________________________________________________________

_________________________________________________________________________________________

LESSON

5-4

Name ________________________________________ Date __________________ Class __________________


101

Applying Operations with Rational Numbers Practice and Problem Solving: A/B

Solve. 1. Four friends equally shared the cost of

supplies for a picnic. The supplies cost $12.40. How much did each pay?

_____________________________________

2. Twenty people are going by van to a movie. Each van seats 8 people. How many vans are needed to take everyone?

_____________________________________

3. Plastic forks come in packs of 6. You need 40 forks for a party. How many packs of forks should you buy?

_____________________________________

4. Kesha spent a total of $9.60 on new shoelaces. Each pair cost $1.20. How many pairs of shoelaces did she buy?

_____________________________________

5. Horses are measured in units called

hands. One inch equals 14

hand. The

average Clydesdale is 17 15

hands tall.

What is this height in inches? In feet?

_____________________________________

6. A banana bread recipe calls for 34

cup

butter. One tablespoon equals 116

cup.

How many tablespoons of butter are needed to make the banana bread?

_____________________________________

7. Cindy works part-time and earns $5.75 an hour. One year she worked 50 weeks and averaged 12.4 hours of work per week. About how much money did she earn that year?

_____________________________________

8. At a gymnastics competition, Joey scored 9.4, 9.7, 9.9, and 9.8. Carlos scored 9.5, 9.2, 9.7, and 9.6. Who had the greater average score? By how many points was his score greater?

_____________________________________

9. A granola recipe calls for 2 13

cups of

almonds. A bag of almonds contains

2 cups. To make 2 12

batches of granola,

Ali buys 5 bags of almonds. How many cups of almonds will he have left over?

_____________________________________

10. At a zoo, 3 pandas eat a total of 181 12

pounds of bamboo shoots each day. The male panda eats 3 times as much as the baby. The female eats twice as much as the baby. How many pounds of bamboo shoots does the female panda eat?

_____________________________________

LESSON

5-5

Name ________________________________________ Date __________________ Class __________________


102

Applying Operations with Rational Numbers Practice and Problem Solving: C

Solve. 1. Sandy makes linen scarves that are

78

of a yard long. How many scarves can

she make from 156 feet of fabric?

_____________________________________

2. A small rug is 36 inches long. Its width is 23

of its length. What is the width of the

rug in feet?

_____________________________________

3. Four friends split equally a lunch bill of $36.96 plus 20% tip. How much did each person pay?

_____________________________________

4. Jade spent $37.60 on groceries. 45

of

that total was spent on vegetables. How much was spent on other items?

_____________________________________

5. In January, Gene watched 5 movies. Their lengths are shown in the table. How many hours did Gene spend

watching movies? _________________

What was the average length of a movie in hours?

_________________

Which movies were longer than the average?________________

6. Derrick’s garden is 18 12

feet long. He plants bulbs 38

of a

foot apart. How many bulbs can Derrick plant in one row?

_________________

Derrick plants three rows of bulbs that cost $0.79 each.

How much does he spend on bulbs? _________________

7. Yin’s cellphone plan costs $30 a month. She used 12.5 hours in May.

What was her cost per minute? _________________

Yin’s average call lasted 3.25 minutes. How much did an average

call cost? _________________

About how many calls did Yin make in May? _________________

LESSON

5-5

Movie Length (min)

A 147.8

B 119.7

C 156.4

D 158.3

E 112.9

Name ________________________________________ Date __________________ Class __________________


103

Applying Operations with Rational Numbers Practice and Problem Solving: D

Solve each problem. The first one has been done for you.

1. A hiking trail is 910

mile long. There are

7 markers evenly posted along the trail to direct hikers. This results in 6 spaces of the same length between the markers. How far apart are the markers?

_____________________________________

2. Tomas is saving $17.00 each week to buy a new sewing machine that costs $175.50. How many weeks will he have to save to have enough money to buy the sewing machine?

_____________________________________

3. Sequins come in packs of 75. Agnes uses 12 sequins on each costume. If she has one pack of sequins, how many costumes can she make?

_____________________________________

4. Jessie pays $2.19 each month for an annual subscription to Sewing magazine. She receives 12 magazines annually. How much does Jessie pay for an annual subscription?

_____________________________________

5. Lisa’s family drove 830.76 miles to visit her grandparents. Lisa calculated that they used 30.1 gallons of gas. How many miles per gallon did the car average?

_____________________________________

6. Jamal spent $6.75 on wire. Wire costs $0.45 per foot. How many feet of wire did Jamal buy?

_____________________________________

7. In England, mass is measured in units called stones. One pound equals 1

14 of a stone. A cat has a mass of

34

stone. What is its mass in pounds?

_____________________________________

8. Dan uses 164

cups of flour to make pita

bread for his family. The recipe calls for 122

cups. How many batches of the pita

bread recipe did he make?

_____________________________________

9. Shari used a total of 67.5 yards of cotton material to make costumes for the play. Each costume used 11.25 yards of cloth. How many costumes did Shari make?

_____________________________________

10. Mike earned $11.76 per hour for working 23.5 hours last week. How much money did Mike earn last week?

_____________________________________

LESSON

5-5

320 of a mile

Name ________________________________________ Date __________________ Class __________________


104

Applying Operations with Rational Numbers Reteach

When a word problem involves fractions or decimals, use these four steps to help you decide which operation to use.

Tanya has 1132

feet of ribbon. To giftwrap boxes, she needs to

cut it into 78

-foot lengths. How many lengths can Tanya cut?

Step 1 Read the problem carefully. What is asked for?

The number of lengths is asked for.

Step 2 Think of a simpler problem that includes only whole numbers.

Tanya has 12 feet of ribbon. She wants to cut it into 2-foot lengths. How many lengths can she cut?

Step 3 How would you solve the simpler problem?

Divide 12 by 2. Tanya can cut 6 lengths.

Step 4 Use the same reasoning with the original problem.

Divide 1132

by 78

.

Tanya can cut 15 lengths.

Tell whether you should multiply or divide. Then solve the problem. 1. Jan has $37.50. Tickets to a concert cost $5.25 each.

How many tickets can Jan buy?

_________________________________________________________________________________________

2. Jon has $45.00. He plans to spend 45

of his money on sports

equipment. How much will he spend?

_________________________________________________________________________________________

3. Ricki has 76.8 feet of cable. She plans to cut it into 7 pieces. How long will each piece be?

_________________________________________________________________________________________

4. Roger has 2 12

cups of butter. A recipe for a loaf of bread requires

34

cup of butter. How many loaves can Roger bake?

_________________________________________________________________________________________

LESSON

5-5

Name ________________________________________ Date __________________ Class __________________


105

Applying Operations with Rational Numbers Reading Strategies: Analyze Information

Word problems contain information that helps you choose which operation to use. Look for clues to help you decide whether to multiply or divide.

Read the problem carefully.

What is given? What are you asked to find?

Given Asked to Find Operation

a whole a fractional part Multiply by the fraction.

a whole and the number of parts the size of the parts Divide.

a whole and the size of a part the number of parts Divide.

Identify the information given and what you are asked to find. Tell whether to multiply or divide. Then solve the problem. 1. A pumpkin weighs 31.3 pounds. It is split into 3 equal pieces.

What does each part weigh?

Given: __________________________________________________________________________________

Find: ____________________________________________________________________________________

Operation and solution: __________________________________________________________________

2. A pumpkin weighs 2223

pounds. What does 16

of it weigh?

Given: __________________________________________________________________________________

Find: ____________________________________________________________________________________


3. A pumpkin weighs 1423

pounds. A grocer wants to cut it into pieces

weighing 122

pounds each. How many pieces can he cut?

Given: __________________________________________________________________________________

Find: ____________________________________________________________________________________


LESSON

5-5

Name ________________________________________ Date __________________ Class __________________


106

Applying Operations with Rational Numbers Success for English Learners

1. How do you know where to place the decimal point in the quotient in Problem 1?

_________________________________________________________________________________________

2. How can you determine if your answer to Problem 2 is correct?

_________________________________________________________________________________________

LESSON

5-5

Name ________________________________________ Date __________________ Class __________________


107

Operations with Decimals Challenge

Divya is cooking dinner. She goes to the market to buy ingredients. The price per pound of different ingredients is shown in the table below.

Ingredient Price (per lb)

Beef $10.65

Onions $2.49

Potatoes $3.29

Tomatoes $8.45

Asparagus $4.99

Divya’s recipe calls for 3.25 pounds of beef, 0.65 pounds of onions, 0.2 pounds of potatoes, 0.15 pounds of tomatoes, and 0.33 pounds of asparagus.

1. How much will Divya pay for all the ingredients in the recipe? Show your work.

_________________________________________________________________________________________

_________________________________________________________________________________________

_________________________________________________________________________________________

_________________________________________________________________________________________

2. Divya decides to make a vegetarian version of the recipe. To do this she replaces beef with 2.5 pounds of chickpeas and 1.75 pounds of lentils. At the market, chickpeas cost $2.49 per pound and lentils cost $3.59 per pound. How much money does Divya save by making the vegetarian version of her meal? Show your work.

_________________________________________________________________________________________

_________________________________________________________________________________________

_________________________________________________________________________________________

_________________________________________________________________________________________

3. Divya also wants to make fruit smoothies for dessert. She purchases a container of rice milk for $3.49, two pounds of strawberries at $4.99 per pound, and six bananas at $0.75 each. How much did Divya spend on dessert? Show your work.

_________________________________________________________________________________________

_________________________________________________________________________________________

MODULE

5

lesson applying gcf and lcm to fraction operations 4-1

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